Computational Algorithms for Improved Synthetic Aperture Radar Image Focusing

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Risto Vehmas

Computational Algorithms for Improved Synthetic Aperture Radar Image Focusing

Julkaisu 1586 • Publication 1586

Tampere 2018

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Tampereen teknillinen yliopisto. Julkaisu 1586 Tampere University of Technology. Publication 1586

Risto Vehmas

Computational Algorithms for Improved Synthetic Aperture Radar Image Focusing

Thesis for the degree of Doctor of Philosophy to be presented with due permission for public examination and criticism in Rakennustalo Building, Auditorium RG202, at Tampere University of Technology, on the 26th of October 2018, at 12 noon.

Tampereen teknillinen yliopisto - Tampere University of Technology Tampere 2018

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Doctoral candidate: Risto Vehmas

Laboratory of Signal Processing

Faculty of Computing and Electrical Engineering Tampere University of Technology

Finland

Supervisor: Prof. Dr. Ari Visa

Laboratory of Signal Processing

Faculty of Computing and Electrical Engineering Tampere University of Technology

Finland

Pre-examiners: Asst. Prof. Dr. Jaan Praks

Department of Electronics and Nanoengineering Aalto University

Finland

Dr. Jan Torgrimsson

Department of Earth and Space Sciences Chalmers University of Technology Sweden

Opponent: Dr.-Ing. Rolf Scheiber

Microwaves and Radar Institute German Aerospace Center Germany

ISBN 978-952-15-4227-5 (printed) ISBN 978-952-15-4236-7 (PDF) ISSN 1459-2045

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Abstract

High-resolution radar imaging is an area undergoing rapid technological and scientific development. Synthetic Aperture Radar (SAR) and Inverse Synthetic Aperture Radar (ISAR) are imaging radars with an ever-increasing number of applications for both civilian and military users. The advancements in phased array radar and digital computing technologies move the trend of this technology towards higher spatial resolution and more advanced imaging modalities. Signal processing algorithm development plays a key role in making full use of these technological developments.

In SAR and ISAR imaging, the image reconstruction process is based on using the relative motion between the radar and the scene. An important part of the signal processing chain is the estimation and compensation of this relative motion. The increased spatial resolution and number of receive channels cause the approximations used to derive conventional algorithms for image reconstruction and motion compensation to break down. This leads to limited applicability and performance limitations in non-ideal operating conditions.

This thesis presents novel research in the areas of data-driven motion compensation and image reconstruction in non-cooperative ISAR and Multichannel Synthetic Aperture Radar (MSAR) imaging. To overcome the limitations of conventional algorithms, this thesis proposes novel algorithms leading to increased estimation performance and image quality. Because a real-time imaging capability is important in many applications, special emphasis is placed on the computational aspects of the algorithms.

For non-cooperative ISAR imaging, the thesis proposes improvements to the range alignment, time window selection, autofocus, time-frequency-based image reconstruction and cross-range scaling procedures. These algorithms are combined into a computationally efficient non-cooperative ISAR imaging algorithm based on mathematical optimization.

The improvements are experimentally validated to reduce the computational burden and significantly increase the image quality under complex target motion dynamics.

Time domain algorithms offer a non-approximated and general way for image reconstruction in both ISAR and MSAR. Previously, their use has been limited by the available computing power. In this thesis, a contrast optimization approach for time domain ISAR imaging is proposed. The algorithm is demonstrated to produce improved imaging performance under the most challenging motion compensation scenarios. The thesis also presents fast time domain algorithms for MSAR. Numerical simulations confirm that the proposed algorithms offer a reasonable compromise between computational speed and image quality metrics.

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Preface

The research presented in this thesis was carried out at the Finnish Defence Research Agency (FDRA) and at the Department of Signal Processing, Tampere University of Technology (TUT), during the years 2015 – 2018. Most of the results were obtained during research projects funded by the Finnish Scientific Advisory Board for Defence (MAanpuolustuksen TIeteellinen NEuvottelukunta, MATINE), whose financial support is gratefully acknowledged.

I would like to express my gratitude to my former superiors, Antti Tuohimaa and Jouko Haapamaa at FDRA, and my current superior, Juha Jylhä at TUT, for providing me the opportunity to carry out this research. Special thanks to all of my colleagues both at FDRA and at TUT for putting up with me and for all the help you have provided during the course of this work. I am also very grateful to my pre-examiners Asst. Prof.

Jaan Praks from Aalto University and Dr. Jan Torgrimsson from Chalmers University of Technology for providing me valuable feedback concerning the thesis. I also want to thank Dr.-Ing. Rolf Scheiber from the Microwaves and Radar Institute of the German Aerospace Center for serving as my opponent in the public examination of this thesis.

Tampere, September 2018 Risto Vehmas

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Acronyms

ATI Along-Track Interferometry

BP Back-Projection

COA Contrast Optimization Autofocus CPI Coherent Processing Interval CSA Chirp Scaling Algorithm DBF Digital Beam-Forming DE Differential Evolution

FFBP Fast Factorized Back-Projection FFT Fast Fourier Transform

FIR Finite Impulse Response GPS Global Positioning System HRWS High-Resolution Wide-Swath ISAR Inverse Synthetic Aperture Radar ISLR Integrated Side-Lobe Ratio LFM Linear Frequency Modulation

MSAR Multichannel Synthetic Aperture Radar MTI Moving Target Indication

RCM Range Cell Migration RCS Radar Cross Section RDA Range-Doppler Algorithm RMA Range Migration Algorithm RSA Range Stacking Algorithm SAR Synthetic Aperture Radar SAS Synthetic Aperture Sonar

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viii Acronyms

SDRS Software Defined Radar Sensor SNR Signal to Noise Ratio

STAP Space Time Adaptive Processing STFT Short Time Fourier Transform TFR Time-Frequency Representation ULA Uniform Linear Array

PGA Phase Gradient Autofocus PSLR Peak to Side-Lobe Ratio PSP Principle of Stationary Phase PTR Point Target Response PRF Pulse Repetition Frequency

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Nomenclature

The various symbols used throughout the thesis are listed here. The following is not an exhaustive list of all the symbols that are used. Special meaning to different symbols is indicated by the use of subscripts and superscripts. Their significance is explained whenever used, if their meaning is not evident from the context. Boldface letters denote vectors or matrices, whose dimensions are specified when they are defined.

Latin alphabet

a basis function coefficient A amplitude envelope function

B frequency bandwidth

c speed of light

D diameter of an Uniform Linear Array (ULA)

d element spacing in an ULA

f basis function

g reflectivity function

gb an estimate ofg

G Fourier transform of g

H loss function in range alignment

I intensity image

k spatial frequency variable L loss function in different contexts M number of slow-time samples

n receiver index

N number of receivers

p sum envelope of ss

P S-method window function

r radial distance (range) variable

ss(r, t) radar signal as a function of range and slow-time

sss(n, r, t) radar signal as a function of receiver number, range, and slow-time

t slow-time variable

T slow-time window length

x, y, z Cartesian spatial coordinates

Greek alphabet

α angle of arrival

β azimuth angle

γ angle coordinate in a local polar coordinate system ix

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x Nomenclature

δ Dirac delta symbol

step length in gradient descent ζ step length in gradient descent

η range shift

θ aspect angle

Θ aspect angle support

λ carrier wavelength

Π rectangle function

ρ range coordinate in a local polar coordinate system τ scaled slow-time variable

φ phase function

Φ Cohen’s kernel function

ψ contrast metric in range alignment Ψ contrast metric in autofocus

Ω S-method window length

Other basic notation

F {·} Fourier transform F⁻¹{·} inverse Fourier transform K {·} keystone formatting operation R {·} rotation operation

R{z} real part of a complex numberz I{z} imaginary part of a complex numberz z^∗ complex conjugate ofz

f∗g convolution betweenf andg f ? g cross-correlation betweenf andg

∠{z} phase angle ofz

Throughout the thesis, the double letter notation for the SAR signal introduced by Raney in [1] is used. In this notation, the first letter insscorresponds to the range (fast-time) variable and the second to the slow-time variable. Lowercase letters correspond to the spatial or temporal domain and uppercase letters denote the signal in the corresponding frequency domain. Correspondingly, we adopt the notationsss for the MSAR signal, where the first letter corresponds to the receiver index.

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List of Publications

This is a compilation thesis based on the following original publications which are referred to as I-VI throughout the text. The publications are reproduced with kind permissions from the publishers.

I R. Vehmas, J. Jylhä, M. Väilä, J. Kylmälä, “A Computationally Feasible Optimiza- tion Approach to Inverse SAR Translational Motion Compensation”,2015 European Radar Conference (EuRAD), Paris, France, pp. 17 – 20, September 2015

II R. Vehmas, J. Jylhä, M. Väilä, A. Visa, “ISAR Imaging of Non-cooperative Objects with Non-uniform Rotational Motion”,2016 IEEE Radar Conference (RadarConf), Philadelphia, PA, USA, pp. 1 – 6, May 2016

III R. Vehmas, J. Jylhä, M. Väilä, A. Visa, “Analysis and Comparison of Multichannel SAR Imaging Algorithms”, 2017 IEEE Radar Conference (RadarConf), Seattle, WA, USA, pp. 340 – 345, May 2017

IV R. Vehmas, J. Jylhä, M. Väilä, J. Vihonen, A. Visa, “Data-Driven Motion Com- pensation Techniques for Noncooperative ISAR Imaging”,IEEE Transactions on Aerospace and Electronic Systems, vol. 54, no. 1, pp. 295 – 314, February 2018 V R. Vehmas, J. Jylhä, “Improving the Estimation Accuracy and Computational

Efficiency of ISAR Range Alignment”,2017 European Radar Conference (EURAD), Nürnberg, Germany, pp. 13 – 16, October 2017

VI R. Vehmas, J. Jylhä, “A Contrast Optimization Algorithm for Back-Projection Image Reconstruction in Noncooperative ISAR Imaging”,EUSAR 2018; 12th European Conference on Synthetic Aperture Radar, Aachen, Germany, pp. 464-469, June 2018

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1 Introduction

As remote sensing and surveillance instruments, radars have several advantages over instruments utilizing shorter wavelengths of the electromagnetic spectrum such as optical, infrared or hyperspectral cameras. The advantages include the capability to measure time delay (which is equivalent to radial distance) very precisely, the fact that radar provides its own illumination, and that it is mostly unaffected by weather conditions such as clouds, smoke or fog [2 – 4]. The rapid advancement of semi-conductor technology has enabled a variety of radar applications for industrial, consumer, and automotive markets [5]. Due to the advancements in digital computing technology, the functionality of the radar is becoming more and more controlled by computer software and digital signal processing algorithms. The term Software Defined Radar Sensor (SDRS) [6] has been coined to illustrate this tendency.

Synthetic Aperture Radar (SAR) and Inverse Synthetic Aperture Radar (ISAR) are remote sensing instruments operating in the microwave region of the electromagnetic spectrum [7 – 9]. They are used for high resolution imaging of the ground (SAR) and moving objects (ISAR). The reconstruction of the radar images is based on utilizing the relative motion between the radar sensor and the imaged scene. The radar system can be mounted on an aircraft or a spacecraft (SAR) or it can monitor moving objects while stationary (ISAR). The relative motion is used to synthesize the effect of a very long antenna producing a very narrow antenna beamwidth corresponding to high spatial resolution. SAR is used for such purposes as area (land or sea) monitoring, creating high precision digital elevation models, military reconnaissance, and disaster monitoring, to name a few [10, 11]. ISAR can be used for air- and maritime surveillance and it provides useful information for non-cooperative target recognition [12]. Multichannel Synthetic Aperture Radar (MSAR) is a form of SAR where multiple receive channels are used to receive the signal. It provides increased performance e.g. for Moving Target Indication (MTI) and space-borne SAR imaging. Modern state-of-the-art SAR and ISAR systems can achieve a spatial resolution of about one decimeter [10, 11].

Fig. 1.1 provides an example of a very high resolution radar image. The radar image in this thesis is defined as a two-dimensional representation of the object’s radar reflectivity in the spatial domain. The magnitude of the reflectivity is represented by the color scale of the image. In the two-dimensional case, the image is the projection of the object’s reflectivity into the ground plane, and thus provides a view of the object as it would be seen from above. To obtain a highly focused image such as the one in Fig. 1.1, the relative motion between the scene and the radar has to be known to within a fraction of the carrier wavelength. In X-band for example, this means a precision of a few millimeters. For airborne SAR, a coarse motion estimate is provided by Global Positioning System (GPS) and inertial measurements units, whereas in non-cooperative ISAR the motion of the object is totally unknown a priori. Data-driven motion compensation is the process

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2 Chapter 1. Introduction

Figure 1.1: The radar image (right) of an object (left) is a two-dimensional representation of the object’s radar reflectivity as a function of spatial position. The color scale of the image represents the magnitude of the reflectivity.

of estimating the unknown motion parameters using only information provided by the received signal itself. This process becomes increasingly important as the desired image resolution becomes higher.

1.1 Motivation

Some of the most important current technological trends governing the development of radar systems are illustrated in Fig. 1.2. Modern radar technology enables the use of larger and larger bandwidths for the radar signal producing a higher range resolution.

The advancements in phased array and digital computing technologies make it possible to increase the number of separate receive channels, all of which are digitized and stored on receive. The increased number of receive channels allows for great flexibility in the digital signal processing. The increased digital computing power that is available makes it possible to process the radar data in novel ways to provide useful information about the surroundings of the radar.

The three important factors of Fig. 1.2 motivate the development of new signal processing algorithms for various radar applications. In this thesis, the considered application is radar imaging with ISAR and MSAR. For ISAR, the increased spatial resolution necessitates the development of new data-driven motion compensation algorithms. This is due to the inherent approximations in well-established algorithms. The approximations start to break down as the resolution becomes higher causing the conventional methods to be performance limited. The increased number of receive channels provides a large number of opportunities for MSAR algorithm development. Moreover, the increased digital computing power enables the use of more generally applicable algorithms for image reconstruction in both MSAR and ISAR.

The performance of MSAR and ISAR imaging algorithms has historically been limited because of the limited available computation power. For most applications, the signal processing should be carried out in real time. The well-established ISAR and MSAR algorithms achieve this by using suitable simplifications, which for example enable the use of Fast Fourier Transform (FFT)-based correlation techniques. With the increasing computing power, it is possible to overcome these limitations providing increased imaging

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1.2. Background 3

Figure 1.2: The important technological trends driving the development of radar systems.

Importantly, these developments enable the radar to obtain more information with better quality about its surrounding environment.

performance. Thus, algorithm development plays a key role in making full use of all the technological developments depicted in Fig. 1.2.

1.2 Background

This thesis focuses on motion compensation and image reconstruction algorithms for ISAR and MSAR. To develop generally applicable algorithms offering the potential for increased imaging performance, it is important to understand the historical perspective and current state-of-the-art of the existing algorithms. Next, an overview of the most prominent existing methods for motion compensation in non-cooperative ISAR and image reconstruction in MSAR imaging is presented.

Data-driven motion compensation algorithms

Since the reconstruction of the radar image requires the knowledge of the relative motion between the radar and the object, non-cooperative ISAR imaging is not possible without some sort of data-driven motion compensation. Motion compensation is normally the first step in the ISAR image reconstruction process. Image quality (e.g. in terms of spatial resolution and image contrast) strongly depends on the accuracy of the motion compensation. Several algorithms for accomplishing motion compensation have been proposed in the literature. In general, data-driven motion compensation in non-cooperative ISAR imaging is a very challenging task. Existing methods are based on suitable approximation and compromises. Many of the state-of-the-art methods rely on several distinct optimization steps, where suitable loss functions are minimized to obtain estimates for the unknown motion parameters [13 – 17].

Motion compensation in non-cooperative ISAR is usually divided in two distinct parts:

translational motion compensation and rotational motion compensation. Translational motion compensation typically consists of two parts also: range alignment and autofocus.

Range alignment provides a coarse estimate for the translational motion. It is usually solved by using a suitable optimization procedure based on quality measures calculated from the amplitude envelopes of the range-compressed signal [14, 15, 18, 19]. In these

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4 Chapter 1. Introduction approaches, a numerical global optimization procedure is used to produce a coarse estimate for the translational motion of the object. For example, a pattern search optimization algorithm was used in [15].

Autofocus refers to the fine translational motion compensation applied after range alignment. It is a well-studied problem in the context of airborne spotlight mode SAR imaging, and a wide variety of different approaches exists. Two widely utilized autofocus approaches are based on Contrast Optimization Autofocus (COA) [13, 17, 20 – 29] and the Phase Gradient Autofocus (PGA) algorithm [30 – 42]. In COA, the loss function is calculated from the image intensities. Examples include the coefficient of variation of the image intensities or amplitudes [13], the sum of the squared intensities [17, 22], and the entropy of the intensity image [14]. The autofocus problem is essentially a local optimization problem [17], and can thus be solved efficiently for example by first order local optimization algorithms [22,25]. PGA uses an iterative procedure based on estimation theory to estimate the unknown phase error. It is computationally more efficient than COA, and is often referred to as the gold-standard autofocus algorithm [31].

A time window optimization procedure can be used to obtain a suitable Coherent Processing Interval (CPI) that is used in the ISAR processing [43]. In this method, the optimal CPI is the one that maximizes the ISAR image contrast. As a solution to rotational motion compensation, a technique called keystone formatting partially compensates for the Range Cell Migration (RCM) [44 – 47] caused by the rotational motion of the object. Additionally, a Time-Frequency Representation (TFR) is used often in the image reconstruction to mitigate the effects of non-linear phase histories [43,45,46,48 – 60]

caused by the object rotation. The former approach was first introduced in [44] in the context of airborne SAR imaging of moving objects. The approach based on TFRs was introduced in [48, 50] in the 1990s and has since matured into a well-established technique. For example, the S-method was demonstrated to surpass the conventional Fourier transform-based range-Doppler approach in [59]. Another way to handle the rotational motion compensation is to isolate strong point-like scatterers from the intensity image and to track their phase progression in the range-compressed signal. This approach is called prominent point processing [8, 61].

An important part of rotational motion compensation is to scale the cross-range dimension of the image into spatial units. This cross-range scaling is a challenging task with no widely accepted general solution [62 – 65]. Existing approaches are based on either estimating high-order phase coefficients from the range-compressed signal [64, 66, 67] or a suitable optimization approach [68, 69]. The optimization approaches rely on dividing the CPI into two or more parts to produce multiple ISAR images. These images are approximated as scaled and rotated versions of each other. Maximizing their correlation can be used to estimate the rotation between them, and the estimated rotation can be used to derive the spatial scale of the image in the cross-range dimension.

An attractive approach to ISAR imaging is to utilize a multistatic system. This refers to a multichannel ISAR system, which can contain multiple transmitters as well as receivers in different spatial positions. The multistatic setup can help in the cross-range scaling problem and in other aspects of the motion compensation as well [70, 71]. The additional information provided by the multiple receiver signals can for example be used to obtain least-squares estimates for the unknown motion parameters of the object [70]. Due to the more complicated imaging geometry, time domain image reconstruction is preferred in the multistatic case [70]. However, the use of time domain image reconstruction algorithms such as [70, 72] in ISAR has been limited, because the data-driven motion compensation

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1.2. Background 5 becomes computationally infeasible for a real-time implementation.

Motion compensation in spotlight mode SAR follows very similar principles than in ISAR. However, it is generally not as difficult, because the radar platform usually carriers inertial sensors and a differential GPS. When combined, these sensors can produce an estimate for the flight track to a precision of a few decimeters [11]. Assuming a high-end navigation system, the precision can be even more accurate. However, especially in the case of light-weight platforms such as unmanned aerial vehicles, the motion errors can be large. This calls for estimation techniques very similar to ISAR processing [73,74]. For space-borne SAR data-driven motion compensation is in general unnecessary, since an almost ideal rectilinear trajectory is possible to achieve. However, atmospheric effects can cause phase errors degrading the image quality of space-borne SAR, and autofocus algorithms can be used compensate them.

Compared to ISAR, the implementation of motion compensation algorithms in SAR is associated with some difficulties. For example, the PGA algorithm [32] only works for spotlight mode imagery and the polar format algorithm [75] in its standard form. This means that it has to be modified for other image reconstruction algorithms [40, 76]. As another example, space-dependent motion compensation [77 – 79] is needed when the scene size is large compared to the standoff distance. Furthermore, the computational cost of the COA techniques becomes a bottleneck for time domain reconstruction algorithms. In part for this reason, computationally efficient autofocus algorithms have been developed for the time domain back-projection image reconstruction algorithm in [80 – 82].

Image reconstruction for MSAR

SAR image reconstruction algorithms can be roughly divided in two classes: fast frequency domain algorithms and exact time-domain algorithms. The Range-Doppler Algorithm (RDA) [83, 84], Chirp Scaling Algorithm (CSA) [1, 77, 78, 85], and Range Migration Algorithm (RMA) [79, 86, 87] belong to the first class, while the second class is based on the back-projection algorithm originating from computer-aided tomography [75,88].

Frequency domain algorithms are based on utilizing the correlation theorem and the FFT algorithm, which significantly reduces the required number of operations in the SAR image reconstruction. For the frequency domain algorithms to be valid, the trajectory of the radar platform needs to be rectilinear and the radar has to be monostatic. On the other hand, time domain algorithms can be used for an arbitrary imaging geometry.

The use of time domain algorithms has been limited because of their large computational cost. Recently, several fast time domain algorithms have been developed [89 – 92]. An important advantage the time domain algorithms have compared to frequency domain algorithms is the fact that the signal does not have to be uniformly sampled in the along- track dimension. Moreover, the trajectory of the radar does not have to be a straight line or a circle (although it simplifies the processing) [93]. Especially with airborne platforms, deviations from the ideal conditions are always present, which result in complicated processing and degraded image quality for fast frequency domain algorithms [79].

The demand for higher spatial resolution and larger area coverage poses contradictory requirements for conventional monostatic SAR systems. The cause of this contradiction is the fact that in the conventional stripmap imaging mode, high spatial resolution in the along-track direction requires a long synthetic aperture, and the unambiguous sampling of the full aperture thus requires a high Pulse Repetition Frequency (PRF) [7,8,94]. The spotlight imaging mode can be used to increase the spatial resolution, but at the cost of

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6 Chapter 1. Introduction reduced area coverage [8]. When conventional waveforms and identical pulses are used to produce a high resolution in the range direction, the PRF sets a limit for the width of the unambiguous range swath. This causes a trade-off between the scene size and spatial resolution that is governed by the imaging geometry and the radar parameters [95,96].

Several techniques that are based on configurations consisting of multiple receiver apertures have been proposed to overcome the above-mentioned dilemma [94 – 103]. The original idea dates back to 1975, when Cutrona [104] proposed a multiple beam system for Synthetic Aperture Sonar (SAS) to overcome the trade-off between cross-range resolution and unambiguous swath width. From the signal processing point of view, the situation in SAS closely resembles that of space-borne SAR (due to the vastly smaller propagation speed of the signal in water). For this reason, techniques for combining high spatial resolution and large area coverage using multiple receiver apertures have been proposed in the recent SAS literature [105 – 111].

Most of the previous work concerning High-Resolution Wide-Swath (HRWS) MSAR imaging has been done in the framework of space-borne SAR systems, beamforming in the azimuth (or cross-range) direction, and fast frequency domain algorithms [96, 112 – 115].

More recently, the computationally efficient monostatic time domain back-projection algorithms [89,90] have been modified for bistatic configurations [91,92]. In addition to the above-mentioned techniques that use beamforming in azimuth, elevation beamforming can also be advantageous especially for space-borne SAR sensors [100,103].

1.3 Objectives

As discussed above, standard solutions to data-driven motion compensation in ISAR and image reconstruction in MSAR are becoming performance limited. This is because they rely on several approximations which no longer hold true under operating conditions required for achieving a very high resolution (≈10 cm). New algorithmic solutions are proposed in this thesis to overcome these limitations. The main objective is

◦ to develop computationally efficient algorithms with increased imaging performance for ISAR motion compensation and MSAR image reconstruction.

The increased imaging performance refers to quantitative image quality metrics, such as the image contrast and resolution. The computational efficiency concerns the real-time imaging capability of the algorithms. The increased imaging performance should not entail a significantly increased computational burden. Also, the increased imaging performance entails an increased overall applicability for the algorithms.

The main objective can be divided into two secondary objectives:

◦ To present a unified optimization framework for non-cooperative ISAR imaging, in which the individual pieces of the motion compensation fit together in a seamless manner; and

◦ to present computationally feasible time domain image reconstruction algorithms for ISAR and MSAR.

The accuracy-efficiency trade-off of the algorithms is an important consideration of this thesis. The aim is to find acceptable trade-offs between image quality and computational

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1.4. Research methods and restrictions 7 efficiency. For the latter secondary objective, the problem consists of determining suitable approximations and compromises, which reduce the computational complexity of the time domain back-projection algorithm for the MSAR system. In addition, the challenge lies in formulating a well-posed and computationally tractable optimization problem for time domain ISAR image reconstruction.

1.4 Research methods and restrictions

The motion compensation and image reconstruction algorithms presented in this thesis have been implemented and tested with MATLAB software developed by the author.

The algorithms have been tested with both simulated and experimental SAR and ISAR data. For non-cooperative ISAR simulations, data were created utilizing the physical optics Radar Cross Section (RCS) simulation tool presented in [116]. The Signature Management Group of the Finnish Defence Research Agency provided the measured turntable data for this thesis. The data from these measurements are used to validate the developed data-driven motion compensation algorithms for ISAR.

In this thesis, only two-dimensional radar imaging is considered. However, some of the methods developed in the thesis can be generalized to work in a three-dimensional setting also. The most important difficulty in the three-dimensional case comes from the additional rotational degrees of freedom. For the MSAR image reconstruction, only an Uniform Linear Array (ULA) antenna was considered. However, the time domain algorithms developed in this thesis can easily be applied to non-uniform array as well.

The restriction was done to provide a meaningful comparison with standard algorithms which are only applicable for an ULA.

1.5 Publications and author’s contribution

The contributions of the included original publications I – VI are summarized below. The research contribution of this thesis can be roughly divided into two categories as depicted in Fig. 1.3. Publications I, II, and IV – VI present improvements to the conventional optimization-based ISAR processing. Time domain image reconstruction algorithms are considered for ISAR in I and VI and for MSAR in III.

Figure 1.3: The research contribution of the thesis can be divided into two categories as depicted.

I In this paper, we propose an optimization approach to ISAR translational motion compensation based on the global range alignment and COA methods. In range

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8 Chapter 1. Introduction alignment, we parametrized the track as a spline polynomial and minimized the mean squared envelope difference of the range-compressed signal. Differential Evolution (DE) was used in solving the global numerical optimization problem.

The COA problem was solved using first order numerical optimization by deriving an expression for the gradient of the loss function. We considered time domain back-projection image reconstruction but the proposed approach is easily extended to other reconstruction techniques as well.

II The paper proposes optimization techniques for enhancing the ISAR image reconstruction process. Specifically, we utilized keystone formatting and time-frequency signal analysis in an optimization framework for the ISAR image reconstruction.

The proposed optimization method estimates the optimal slow-time window by minimizing a loss function depending on the contrast of the sum envelope of the range compressed signal. The optimal kernel function for the Cohen’s class TFR was determined by maximizing the contrast of the intensity normalized ISAR image.

Furthermore, the time-frequency reassignment method was utilized to enhance the contrast of the ISAR image. In the numerical results, the proposed optimization steps were shown to increase the image contrast.

III The paper formulates the SAR image reconstruction for a multichannel system in its most general form. We utilized Digital Beam-Forming (DBF) and the phase center approximation to develop a fast time domain (Fast Factorized Back-Projection (FFBP)) algorithm for MSAR. We presented two FFBP implementations for MSAR and performed a comparative study between MSAR imaging algorithms. Considering the numerical simulation results, the proposed reconstruction algorithms were significantly faster than exact time domain image reconstruction with essentially the same achieved image quality metrics. The proposed time domain MSAR algorithms provide an alternative for conventional frequency domain image reconstruction, especially in cases where the operating conditions are not ideal for the frequency domain algorithms.

IV The paper describes how optimization can be used in every part of data-driven motion compensation and ISAR image reconstruction. Several improvements were proposed to the range alignment, time-window selection, autofocus, time-frequency- based image reconstruction and cross-range scaling procedures. The range alignment method was enhanced by combining previously suggested loss functions and utilizing first order numerical optimization. In the time window optimization process, we proposed performing autofocus and keystone formatting before evaluating the contrast of the ISAR image. In COA, we utilized a computationally efficient optimization procedure by deriving an expression for the second order partial derivatives of the loss function. For the time-frequency based imaging approach, we chose the optimal kernel for the TFR based on the image contrast. Finally, the rotation correlation and polar mapping methods were combined to solve the cross- range scaling problem in a straightforward manner. By combining all the proposed improvements, a computationally efficient ISAR algorithm was demonstrated. It improved the imaging performance in terms of the image contrast by 50 percent at best and 28 percent on average under complicated target motion dynamics.

V The paper presents methods for improving both the estimation performance and computational efficiency of ISAR range alignment algorithms using mathematical optimization. We proposed new loss functions and a new optimization method

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1.6. Outline 9 based on the first and second order partial derivatives of the loss function. The new loss functions were introduced to obtain better performance under significant target rotation and very high range resolution. In numerical experiments with very high resolution X-band ISAR data the proposed loss functions were shown to increase the estimation performance as much as 35 percent. Moreover, the proposed numerical optimization method reduced the computational cost of optimization-based range alignment by an order of magnitude.

VI The paper considers ISAR imaging of non-cooperative objects exhibiting challenging unknown motions using the time domain back-projection algorithm. We proposed a contrast optimization approach for the data-driven motion compensation problem, in which a novel approach using the gradient of the loss function was utilized in the optimization. The translation and the rotation of the object were estimated simultaneously using first order numerical optimization. The algorithm utilized the methods of IV to produce initial guesses for the unknown motion parameters to speed up its convergence. In numerical experiments, the proposed optimization approach was able to estimate highly non-linear translational and rotational motions producing well-focused ISAR images.

In all of the included publications, the author has been responsible for deriving the theoretical results, developing and implementing the computational algorithms, performing the numerical experiments, analyzing the results, and writing the manuscripts. The co- authors provided help in designing the numerical experiments, interpreting the results, and provided valuable feedback and suggestions to help improve the manuscripts. The experimental data used in the studies has been provided by the Signature Management Group of the Finnish Defence Research Agency. The comments and suggestions made by the anonymous peer-reviewers also helped improve the quality of the manuscripts.

1.6 Outline

This thesis is divided into 5 chapters. In Chapter 1, the background and motivation for the study are given. They are followed by the objectives, research methods, restrictions and contributions of the thesis. Chapter 2 introduces the basic principles of ISAR and MSAR image reconstruction and motion compensation algorithms. The research contribution of the thesis is summarized in Chapters 3 and 4. Chapter 3 presents the novel algorithms developed for optimization-based data-driven motion compensation in non-cooperative ISAR imaging. Chapter 4 presents the results related to the developed time domain image reconstruction algorithms for both ISAR and MSAR. The achieved results are discussed and critically evaluated in Chapters 3 and 4. Chapter 5 summarizes the most important findings and concludes the thesis.

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2 Principles of synthetic aperture imaging and motion compensation

To motivate and understand the algorithms proposed in this thesis, it is essential to have a suitable model for the radar signals of interest. Furthermore, the signal models and their underlying assumptions clarify the significance of different parts of the proposed ISAR and MSAR algorithms. Section 2.1 describes the basic ISAR signal model used for two- dimensional imaging. The concepts of translational and rotational motion compensation are defined and illustrated in Sections 2.2 and 2.3, respectively. Conventional methods for solving them are described. The principles of time domain ISAR image reconstruction are presented in Section 2.4. Section 2.5 concludes the theoretical foundation by introducing the MSAR signal model and the conventional image reconstruction process.

2.1 Basic ISAR principles

Fig. 2.1 illustrates the basic ISAR geometry. The basic ISAR signal model is extensively discussed e.g. in [14, 18, 43, 50, 55, 60, 64, 67, 68]. The model is appropriate for a two- dimensional imaging geometry where the non-cooperative object is constrained to move on a plane. The primed coordinate system, whose origin is assumed to be located in the object’s center of mass, is rigidly attached to the object. The spatial degrees of freedom for each instant of slow-time includer₀ (the location of object center of mass) and the orientation of the primed coordinate system (the heading of the object). Slow-time tis the time variable that remains constant during the formation of a single range profile and increases from one range profile to the next. Importantly, the aspect angleθ changes as the object moves in a suitable way. Consequently, the range R_p(t) =||R_p(t)|| between every spatial position on the object and the radar changes in a unique way as a function of slow-time.

In most practical ISAR scenarios, we are interested in imaging vehicles such as cars, ships, or airplanes. The distance between the radar and the object is usually several kilometers, whereas the dimensions of the imaged objects are much smaller. Thus, we have kr0k = r₀ krpk =r_p, which means that the object is in the far-field. In IV, we showed how this argument can be used to derive an expression for the distanceR_p between an arbitrary point on the object and the radar. The result is

R_p(t)≈r₀(t) +x_pcosθ(t) +y_psinθ(t), (2.1) wherex_p=r_psinθ_pandy_p=r_pcosθ_p. This is a key result from the motion compensation perspective, because the motion can be divided into two parts. The first part (r₀) is called the translational motion. It is the same for every image position (spatially invariant) and

11

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12 Chapter 2. Principles of synthetic aperture imaging and motion compensation

Figure 2.1: Basic ISAR geometry for a vehicle moving on the ground. As the object moves, the distanceRpbetween an arbitrary point on the object and the radar changes uniquely.

thus is not useful for the image reconstruction. The second part is the rotational motion, which produces the unique phase histories for different image positions (spatially variant) required for obtaining the cross-range resolution.

Adopting the usual start-stop approximation, assuming a constant wave propagation speed, and using the principle of superposition, the Point Target Response (PTR) of the system as a function of radial distancer and slow-timet after range compression and quadrature demodulation is

ss_p(r, t) = sinc 2B

c (r−R_p(t))

e⁻ⁱ^4π^λc^R^p^(t), (2.2) whereλ_c is the carrier wavelength,B is the temporal frequency bandwidth of the signal, andc is the propagation speed of the radio wave. The result (2.2) can be derived by

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2.2. Translational motion compensation 13 assuming that the transmitted signal is band-limited [117]. The Fourier transform of (2.2) with respect to the radial distance (range) variableris

Ssp(kr, t) = Π k_rc

2πB

e^−i2(k^r^+k^c^)[r⁰^(t)+x^p^cos^θ(t)+y^p^sin^θ(t)], (2.3) where kr is the range spatial frequency variable,kc = (2π)/λc and Π is the rectangle function. By denoting the reflectivity function of the object as g, we can express the output at the radar receiver as a convolution between the PTR and g. By using the convolution theorem, this yields the result

Ss(k_r, t) =e^−i2(k^r^+k^c^)r⁰^(t)Π k_rc

2πB

G(2k_rcosθ(t),2k_rsinθ(t)) (2.4) in the spatial frequency domain. In (2.4), G(kx, ky) =F_x→k_x

F_y→k_y{g(x, y)} is the two-dimensional Fourier transform of the reflectivity functiong.

The expression in (2.4) is a key result for both ISAR and spotlight mode SAR. It can be interpreted as follows: The range-compressed radar signal is a series of phase- modulated slices of the two-dimensional Fourier transform of the reflectivity function g. The rectangle function in (2.4) represents the fact that the signal is essentially band- limited inkr. Naturally, the signal also has a limited support in slow-time (and thus inθ), which is omitted in (2.4) for simplicity. Consequently, the slices actually span an annular sector in the two-dimensional spatial frequency domain. In the context of spotlight-mode SAR, the result (2.4) was originally derived by using the projection slice theorem [75,118].

In non-cooperative ISAR, the difficulty arises from the fact that both r₀ and θ are unknown a priori. To reconstruct a properly focused image, their values as a function of slow-time have to be estimated somehow. As is evident from (2.4), r₀ is a nuisance that does not provide any useful information for the image reconstruction. Its estimation and compensation is referred to as translational motion compensation. Correspondingly, the estimation and compensation ofθis referred to as rotational motion compensation.

2.2 Translational motion compensation

The purpose of translational motion compensation is to estimater0and to compensate for it prior to the ISAR image reconstruction. As indicated by (2.4), the compensation can be achieved in the range spatial frequency-slow-time (k_r, t) domain by multiplying the signal with the complex conjugate of the exponential term on the right hand side of the equation. The space-invariance of the translational motion makes its estimation and compensation more simple than the space-variant rotational motion compensation.

The space-invariance can be exploited in the motion compensation algorithms. For example, estimates from several range bins can be averaged in the estimation process to achieve improved estimation performance and the compensation can be achieved by a very computationally efficient complex multiplication in the (kr, t) domain.

Range alignment

Traditionally, translational motion compensation is performed in two distinct parts:

range alignment and autofocus. The purpose of range alignment is to compensate for translational motions larger in magnitude than the range resolution. The estimation process is usually based on quality measures calculated from the shifted amplitude

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14 Chapter 2. Principles of synthetic aperture imaging and motion compensation envelopes of the range-compressed signal [14 – 16, 19, 119]. According to the Fourier shift property, the linear phase term (as a function ofkr) corresponds to a shift in the range domain. By compensating for the translational phase term in the frequency domain, the range profiles are shifted such that they are “aligned”. This is where the term range alignment originates from.

The range alignment algorithms relying on mathematical optimization are based on the minimum entropy method [14] and global range alignment [15, 16]. These methods are optimally suited for small rotational angles and coarse range resolutions, because they approximate that the shape of the amplitude envelope of the range profiles remains constant during the CPI. The basic formulation of these methods is based on minimizing a loss functionL, which is a function of the unknown range shiftsη∈R^M. The value of Lquantitatively measures the quality of the range alignment. Thus, the range alignment problem is equivalent to solving

η^?= arg minL(η). (2.5)

The global range alignment method is based on quality measures calculated from the sum envelope, which is defined as

p(r;η) =

M−1

X

m=0

|ss(r+η_m, t_m)|, (2.6) whereη_m is themth component ofη. Using the Fourier shift property, the range-shifted range-compressed signal can be obtained as

ss(r+η_m, t_m) =F_k⁻¹

r→r

e^i2k^r^η^mSs(r, t_m) . (2.7) The loss functions of the global range alignment can be expressed as

L(η) =Z ∞

−∞

ψ(p(r,η))dr, (2.8)

whereψ:R→Ris a suitable function used to quantify the quality of the sum envelope.

For example,ψ(p) =−p²is used to measure the contrast andψ(p) =−plogpthe entropy of the sum envelope. Another loss function used in the literature is the mean squared envelope difference [119], which can be expressed as

L(η) =Z ∞

−∞

M−2

X

m=0

(|ss(r+ηm+1, tm+1)| − |ss(r+ηm, tm)|)²dr. (2.9) The advantage of (2.9) compared to (2.8) is that scatterer RCM does not degrade its performance as significantly, because only adjacent time samples (range profiles) are subtracted in the loss function. As a downside, this may cause error accumulation.

As we have shown in V, the loss functions of the type (2.8) are not convex. Consequently, solving the range alignment requires global numerical optimization, which is computationally very demanding. Moreover, heuristic numerical global optimization methods do not guarantee that the optimal solution is even found. The minimum entropy method [14]

circumvents this difficulty by using only two adjacent slow-time samples (range profiles) in the definition of the sum envelope (2.6). The method proceeds to solve theseM−2

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2.2. Translational motion compensation 15

Figure 2.2: The intensity of the range-compressed signal before (left) and after (right) range alignment. The translational motion of the object shifts the response in range and the range alignment compensates for this slow-time-dependent shift.

one-dimensional optimization problems separately. Rapid changes in the target reflectivity as well as the possibility of error accumulation cause problems for this approach. Global range alignment [15, 16] avoids error accumulation and is more robust against target scintillation. However, it comes with the cost of numerical global optimization in a multi-dimensional search space. In [15], a pattern search algorithm and a polynomial model for the range shifts were used to produce a computationally realizable algorithm.

Fig. 2.2 illustrates the effects of translational motion and the result of range alignment in ISAR processing. The signal in question is measured radar data of the object depicted in Fig. 1.1. As indicated by (2.4), the translation of the object manifests itself as a slow-time dependent range shift of the object’s radar response. The range alignment eliminates these shifts and aligns the range profiles as a function of slow-time.

Autofocus

After range alignment is performed, a slow-time-dependent phase error caused by the residual translational motion remains in the signal. In conventional ISAR processing, it is assumed that the range alignment has compensated for the translational motions that are larger in magnitude than the range resolution. Thus, taking the inverse Fourier transform of (2.4) after applying the range alignment phase correction results in

ss(r, t)≈e^iφ^e^(t) Z Z ∞

−∞

g(x_p, y_p)ss_p(r, t)dx_pdy_p, (2.10) whereφ_eis the residual phase error. If this holds true, the residual translational motion can be compensated for by applying a one-dimensional phase correction in the range-slow- time (r, t) domain. The estimation and compensation of this spatially invariant phase term is a widely studied problem in spotlight mode SAR [8,17,22,25,35,42,118,120,121]. This computer-automated image focusing procedure is called autofocus. The most prominent autofocus algorithms are based either on the PGA algorithm [30,32] or COA (also known as sharpness maximization and entropy minimization in the literature) [13,14, 17,22,25].

To arrive at the PGA and COA algorithms, it is useful to consider the simplest algorithm for image reconstruction in non-cooperative ISAR. Using the far-field and small angle

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16 Chapter 2. Principles of synthetic aperture imaging and motion compensation approximations, we can approximate the PTR as

ssp(x, θ) = sinc2B

c (x−xp)

e^−i2k^c^(x^p^+y^p^θ). (2.11) In (2.11),thas been replaced by the aspect angleθ(without any loss of generality, because the PTR depends ontonly throughθ) and the radial distance variableris approximated by the Cartesian coordinatex. This approximation is accurate when the object dimension is very small compared to the standoff distancer0. Substituting (2.11) into (2.10) results in

ss(x, θ) =e^iφ(θ)e^−i2k^c^xF2k_cy→θ

g(x, y)∗xsinc2B

c (x−xp)

, (2.12) where the notation∗_xstands for convolution with respect to x. Thus, to reconstruct the image (an estimate ofg), an inverse Fourier transform with respect toθsuffices provided that the phase error φe is compensated for prior to it. This simple approach based on a one-dimensional inverse Fourier transform is called the range-Doppler algorithm in the ISAR literature [122]. There also exists an algorithm called range-Doppler for stripmap mode SAR image reconstruction [83], which is not to be confused with this ISAR algorithm.

To produce a focused ISAR image of the object, the phase errorφein (2.12) needs to be compensated for prior to the image reconstruction. PGA estimatesφeby isolating the phase histories of the strongest scatterers in each range bin [30, 32, 118]. The isolation is achieved by a windowing operation in the spatial (x, y) domain, after which the data is Fourier-transformed back to the (x, θ) domain. The algorithm obtains an estimateφbefor the phase error based on evaluating

φbe(θ) =Z θ

−∞∠ Z ∞

−∞

exp i∠

∂ss(x, θ⁰)

∂θ⁰

dx

dθ⁰. (2.13) In practice, the derivative of the range-compressed signal in (2.13) can be estimated by a simple complex conjugate multiplication procedure followed by the integration over range bins. Because the isolation of strong point-like targets is not perfect and depends on the initial image focus, the PGA algorithm is usually applied iteratively. In spotlight mode SAR imaging, it has been shown to exhibit excellent performance in estimating high-frequency phase errors over a wide variety of scene types [8, 32].

Another widely utilized approach to autofocus in both SAR and ISAR is COA. As its name suggests, it estimates the phase errors by maximizing the image contrast. The method is conceptually very simple and can be expressed as a single simple equation

φ^?= arg min

φ∈[−π,π]^M

L(φ), (2.14)

whereφm=φbe(tm). The loss functionL is calculated from the image intensities as L(φ) =Z Z ∞

−∞

Ψ(I(φ;x, y))dxdy, (2.15) whereI(φ;x, y) =|sS(φ;x, y)|² is the image intensity and

sS(φ;x, y) =F_θ⁻¹

m→y

ss(x, θ_m)e^−iφ^m . (2.16)

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2.3. Rotational motion compensation 17

(a)SAR image before autofocus. (b)SAR image after autofocus.

Figure 2.3: Illustration of the effects of uncompensated motion errors and autofocus in SAR images. Phase errors cause the image to be out of focus in (a), while autofocus sharpens the image in (b) by correcting the phase errors in the data.

The function Ψ is sometimes called the contrast metric [17]. Common choices for it include the power law Ψ(I) =−I^a, wherea∈Rand usuallya >1, and the entropy Ψ(I) =−IlnI. The earliest COA methods were based on using a parametric model for the phase errors and applying black-box numerical optimization techniques [13, 14]. The drawback of these methods was the high computational burden caused by the optimization algorithm.

Fienup introduced a computationally efficient first order optimization procedure for COA in [23,25] by deriving expressions for the gradient under the conventional spotlight SAR paradigm. Since then, COA has matured in a well-established approach in both SAR and ISAR autofocus. COA works generally well for objects and scenes containing bright point-like targets.

Fig. 2.3 illustrates the image defocusing caused by phase errors and the subsequent image focusing using an autofocus algorithm. The spotlight SAR data-set used for this demonstration is from the Gotcha challenge project [123]. The scene consists of a parking lot and the road surrounding it. The resolution of the image is approximately one foot in both directions. The uncompensated phase errors in the data cause the SAR image to be severely defocused, as seen form Fig. 2.3a. The responses of the scatterers are smeared in the cross-range direction of the image, which causes a loss of resolution and image contrast. The defocusing effect is the same throughout the image (spatially invariant).

Applying the PGA algorithm to the image data in Fig. 2.3a produces the well-focused SAR image in Fig. 2.3b as a result.

2.3 Rotational motion compensation

Let us consider the signal model (2.4) assuming that the phase term caused by the translational motion of the object has been completely compensated for. To obtain an estimate for the object reflectivity function, the aspect angle θhas to be known. The purpose of rotational motion compensation is to estimate θ and to compensate for its effects in the image reconstruction. In general, rotational motion compensation is a more difficult task than translational motion compensation. The reason for this is the fact that it causes spatially variant effects in the image.

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18 Chapter 2. Principles of synthetic aperture imaging and motion compensation

Keystone formatting

To consider the need for rotational motion compensation, it is illustrative to express the signal model as

ss(r, θ) =Z Z ∞

−∞

g(x, y)sinc2B

c (r−xcosθ−ysinθ)

e^−i2k^c^(x^cos^θ+y^sin^θ)dxdy. (2.17) Two important observations can be made from (2.17). Firstly, the amplitude envelope of each scatterer follows the curveR =xcosθ+ysinθ causing spatially variant RCM.

Secondly, the phase term inside the integral is not linear inθ. Essentially, this means that the simple range-Doppler approach based on (2.12) is not adequate to produce a focused image. In the ISAR literature, the first problem has been partially solved by utilizing an approach called keystone formatting [44 – 46]. The second problem can be handled by replacing the Fourier transform with a suitable TFR [48 – 50,52,54, 55].

The keystone formatting operation is derived by considering the phase of the signal in the (kr, t) domain. Using (2.3), this can be expressed as

φ_p(k_r, θ) = 2 (k_r+k_c)R_p(θ), (2.18) whereRp(θ) =xpcosθ+ypsinθ (after translational motion compensation). This divides into the carrier termkcRp(θ) and the termkrRp(θ), which determines the range location of the scatterer in the range-compressed signal. RCM occurs if 2Rp changes more than the range resolution during the CPI. Substituting the Maclaurin series expansion ofRp

into (2.18) results in

φp(kr, t) = 2(kr+kc)(Rp(0) + ˙Rp(0)θ+R¨p(0)

2! θ²+O(θ³)), (2.19) where ˙R=dR/dθ. Examining (2.19) reveals that the RCM caused by any single term in the Taylor expansion can be removed by a suitable change of variables. Most commonly, the linear RCM is removed by making the change of variablesθ→τ according to

θ= k_cτ

kr+kc

. (2.20)

In practice, the change of variables in (2.20) is carried out by using an Finite Impulse Response (FIR) filtering-based resampling of the signal. Notably, this operation removes the RCM regardless of ˙Rp, which means that it simultaneously compensates for the linear RCM of all scatterers.

Time-frequency-based image reconstruction

To avoid image defocusing caused by the rotational motion, the non-linear phase histories of the echoes scattered from different parts of the object need to be taken into account.

An effective way of doing this without having to explicitly estimate the aspect angle θis to replace the inverse Fourier transform used in the range-Doppler approach by a high-resolution TFR. This is a well-established technique in non-cooperative ISAR and several TFRs have been utilized for this purpose in the literature [49 – 51,54 – 60]. Almost all of these methods are based on using a quadratic TFR. Quadratic in this context means that the TFRs are based on a multiplicative comparison of the signal with itself.

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2.3. Rotational motion compensation 19

(a)ISAR image before rotational motion compensation.

(b) ISAR image after rotational motion compensation.

Figure 2.4: Illustration of the effects of uncompensated rotational motion and rotational motion compensation in ISAR images. Uncompensated non-uniform rotation causes significant spatially variant blurring in (a), which is corrected by the rotational motion compensation algorithm applied in (b).

Quadratic TFRs can be conveniently formulated using the Cohen’s class [124]. In this formulation, the TFR and thus the ISAR image reconstruction can be expressed as

bg(x, y, θ) =Z Z ∞

−∞

Φ(θ−µ, y−ν)W D(x, y, θ)dµdν, (2.21) where

W D(x, y, θ) =F_ν→yn

ss(x, θ+ν

2)ss^∗(x, θ−ν 2)o

(2.22) is the Wigner-Ville representation of the signal and Φ is the smoothing kernel function defining the particular distribution. As is evident from (2.21), this method produces a time series of ISAR images of the object. Either the entire time series or a selected subset of images can be exploited in the subsequent image exploitation steps. Although effective in ISAR, the choice of the kernel function is critical for the image reconstruction to be successful. Existing literature provides examples of specific choices but lacks the tools for automatic and optimal choices for the kernel function.

Fig. 2.4 illustrates the effects of uncompensated rotational motion and the benefits of successfully applying rotational motion compensation. The dataset used in this example are is a simulated response of a car in S-band. In this example, the range resolution is very high (≈10 cm) and the car rotates almost 30 degrees during the CPI. This causes significant RCM and non-linearity in the phase history of the signal. Without proper compensation, the rotational motion causes significant spatially variant defocusing, as seen in Fig. 2.4a. This defocusing not only decreases the resolution and image contrast, but makes it very hard to determine the shape of the object. After applying keystone formatting and using the S-method as the TFR in (2.21), the ISAR image in Fig. 2.4b is obtained. This time-snapshot is the one where the image contrast is the highest of all the time frames. The spatially variant defocusing is removed and the shape of the object is clearly distinguishable from Fig. 2.4b.

Computational Algorithms for Improved Synthetic Aperture Radar Image Focusing

Risto Vehmas

Computational Algorithms for Improved Synthetic Aperture Radar Image Focusing

Julkaisu 1586 • Publication 1586

Tampere 2018

Risto Vehmas

Computational Algorithms for Improved Synthetic Aperture Radar Image Focusing

Abstract

Preface

Contents

Acronyms

Nomenclature

List of Publications

1 Introduction

1.1 Motivation

1.2 Background

1.3 Objectives

1.4 Research methods and restrictions

1.5 Publications and author’s contribution

1.6 Outline

2 Principles of synthetic aperture imaging and motion compensation

2.1 Basic ISAR principles

2.2 Translational motion compensation

2.3 Rotational motion compensation